The field of the invention relates generally to non-linear resonators. Resonant-based sensing has long been exploited to gauge micromechanical system parameters such as mass and spring constant. Gravimetric sensors and mass flow meters that measure the mass change and the stress sensors that link to suspension spring constant changes all exemplify this approach. As harmonic resonators suffer from a direct link between amplitude and frequency noise that degrades sensitivity, it would be advantageous to use a class of nonlinear parametric resonators for developing an extremely sensitive mechanism to reveal tiny changes in resonance characteristics. This improved sensitivity stems from an instantaneous “jump” transition from near-zero to high displacement in parametric resonators, yielding high ratio of resonant amplitude to the minimum detectable amplitude signal.
The harmonic and parametric resonators differ in the way the resonance is excited. The harmonic resonator has a fixed spring constant, with an external drive that is sinusoidal in time t and invariant in displacement x. The parametric resonator is driven by a nonlinear electrostatic drive at an electrical frequency that creates a displacement-dependent spring constant keff. Parametric resonance conditions are satisfied when the periodic variation of keff is at twice the natural mechanical resonance frequency or at other integer multiples fraction of twice the natural mechanical resonance frequency (i.e., 2ωr/n, where ωr is the natural mechanical resonant frequency and n is an integer larger or equal to 1).
A microelectromechanical parametric resonator operates in two states—one “off” state that has zero motion and the other “on” state whose amplitude increases exponentially over time. In the “off” state, the resonator is nominally perfectly balanced at rest. However, because of the existence of thermomechanical noise or with the existence of external disturbances, a very small but non-zero noise displacement typically on the order of femtometers to nanometers is present, depending on the level of noise inherent in the system. With non-zero displacement and application of a parametric drive voltage under appropriate pump frequencies where the parametric resonance condition is satisfied, the mass-spring-damper system periodically increases (hardens) or decreases (softens) its effective spring force, resulting in the mass being pumped into parametric resonance.
An instantaneous bifurcation “jump” is seen from a near-zero “off” state (point A) to a non-zero high amplitude (point B) “on” state in the representative steady-state amplitude versus frequency characteristic in FIG. 1. This sharp transition from A to B is called “bifurcation”. Quality factor (Q), which is inversely related to energy loss per cycle, is an important metric used to evaluate the performance of a resonator. Lower loss generally gives rise to a sharper amplitude peak in the frequency spectrum. The infinitely steep slope on the bifurcation edge effectively enhances Q-factor. This intriguing property promotes many ultra-sensitive sensor applications such as mass flow rate sensors, gravimetric sensors, magnetometers and strain sensors, and is used in noise-squeezing applications for parametric amplification.
An analogous bifurcation response occurs in a micromechanical Duffing resonator, where the bending of the resonance amplitude curve is due to the mechanical nonlinearity. The bending leads to multi-valued solutions for one particular frequency, which results in “bi-stability” in frequency response.
To enable parametric resonance and/or Duffing resonance to be used in making an ultra-sensitive mass sensor or strain gauge, a drive scheme is needed to excite the parametric resonance and control the servo such that the controlled setpoint along the bifurcation “jump” is maintained. Bifurcation-based control approach can be realized both open-loop and closed-loop. The former is slow, sometimes taking up to minutes since the parametric drive frequency is swept with fine increments until a jump in amplitude is observed, and is impractical for most practical (real-time) sensing applications. The closed-loop scheme in principle can servo along the jump point (for example, the point along A-B in FIG. 1) enabling rapid readout of the bifurcation amplitude and/or frequency. There are two practical limits to this approach. One is due to the hysteresis (jumps can occur at A-B or S-P depending on the direction of the frequency sweep), which inhibits the amplitude from readily moving back to its original state once the amplitude has transitioned across the bifurcation. The other limitation originates from a slow-time manifold that the resonator state has to travel in order to transit between “on” and “off” states. This slow-time manifold on the system trajectory is relatively long (on the order of seconds), inhibiting stable servo operation when linear feedback control approaches are applied.
One analog bifurcation control approach takes the phase variance as the input of the controller. These controllers are generally complex and difficult for on-chip implementation. It also produces a relatively small amplitude on the order of 100 pm. The advantage of the controller of the present invention is its practical feasibility for on-chip circuit implementation and its ability to servo at relatively large amplitude (typically about four orders of magnitude larger in the displacement amplitude than the 100 pm amplitude) along the sharp jump of the bifurcation without altering the plant.